Hockey-stick identity

The hockey-stick identity states that a diagonal sum of binomial coefficients equals a single binomial coefficient: \(\sum_{j=r}^{n}\binom{j}{r}=\binom{n+1}{r+1}\).

Pascal's rule: Pascal's rule expresses each interior binomial coefficient as the sum of the two coefficients above it. Repeatedly applying this recurrence makes adjacent terms cancel in a telescoping proof.
Pascal's triangle: Pascal's triangle is an arrangement of binomial coefficients in which each interior entry is the sum of the two above it. The identity describes a diagonal run of entries in this triangle.
Subset: A subset is a set whose elements all belong to a specified larger set. Counting subsets by their largest element yields the identity's summation.
Vandermonde's identity: Vandermonde's identity expresses a sum of products of binomial coefficients as a single binomial coefficient. It combines two coefficient sequences rather than summing one diagonal.
Telescoping sum: A telescoping sum is a finite sum whose successive terms cancel, leaving only boundary terms. The hockey-stick sum telescopes after each term is rewritten using Pascal's rule.
Binomial theorem: The binomial theorem expands a power of a sum using binomial coefficients. It connects the coefficients in the identity to polynomial expansions.
Order statistic: An order statistic is a value selected by rank after observations are arranged in order. Counting possible ranks connects hockey-stick sums to distributions of sample order statistics.
Binomial inversion: Binomial inversion converts between two sequences through alternating binomial-coefficient sums. Its alternating transform differs from the hockey-stick identity's unweighted diagonal sum.
Combinatorial proof: A combinatorial proof establishes an identity by counting the same finite set in two ways. The identity follows by counting subsets according to their largest selected element.
Finite sum: A finite sum is the result of adding a specified finite collection of terms. The identity evaluates a particular finite sum in closed form.

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